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On error control in the element-free Galerkin method. (English) Zbl 1245.65161
Summary: We investigate discretisation error control in the element-free Galerkin method (EFGM) highlighting the differences from the finite element method (FEM). We demonstrate that the (now) conventional procedures for error analysis used in the finite element method require careful application in the EFGM, otherwise competing sources of error work against each other. Examples are provided of previous works in which adopting an FEM-based approach leads to dubious refinements. The discretisation error is here split into contributions arising from an inadequate number of degrees of freedom $$e_{h}$$, and from an inadequate basis $$e_{p}$$. Numerical studies given in this paper show that for the EFGM the error cannot be easily split into these component parts. Furthermore, we note that arbitrarily setting the size of nodal supports (as is commonly proposed in many papers) causes severe difficulties in terms of error control and solution accuracy. While no solutions to this problem are presented in this paper it is important to highlight these difficulties in applying an approach to errors from the FEM in the EFGM. While numerical tests are performed only for the EFGM, the conclusions are applicable to other meshless methods based on the concept of nodal support.

##### MSC:
 65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs 65N50 Mesh generation, refinement, and adaptive methods for boundary value problems involving PDEs